K-statistic
In statistics, a k-statistic is a statistic constructed from a sample to estimate a cumulant of the underlying population distribution. For a sample of independent and identically distributed random variables, the rth k-statistic is the unique symmetric unbiased estimator of the rth cumulant and has minimum variance among unbiased estimators.[1]
The first k-statistic is the sample mean, which estimates the first cumulant. The second k-statistic is the usual unbiased sample variance, which estimates the second cumulant.[1] Higher-order k-statistics provide corresponding unbiased estimators of higher-order cumulants.
History
[edit]K-statistics were developed by Ronald Fisher in his work on the moments and product moments of sampling distributions.[2]
Applications
[edit]Multivariate k-statistics can be used to estimate joint cumulants. In signal processing, they have been applied to the unbiased estimation of polyspectra (higher-order spectra), including the bispectrum and trispectrum.[3]
See also
[edit]References
[edit]- 1 2 Di Nardo, Elvira; Guarino, Giuseppe (2022). "kStatistics: Unbiased Estimates of Joint Cumulant Products from the Multivariate Faà Di Bruno's Formula". The R Journal. 14 (2): 208–228. doi:10.32614/RJ-2022-033. ISSN 2073-4859.
- ↑ Fisher, R. A. (1930). "Moments and Product Moments of Sampling Distributions". Proceedings of the London Mathematical Society. 2. 30 (1): 199–238. doi:10.1112/plms/s2-30.1.199.
- ↑ Sifft, Markus; Ghorbanietemad, Armin; Wagner, Fabian; Hägele, Daniel (2026). "Correct estimation of higher-order spectra: From theoretical challenges to practical multi-channel implementation in SignalSnap". Digital Signal Processing. 173 105893. doi:10.1016/j.dsp.2026.105893.
External links
[edit]- k-Statistic at Wolfram MathWorld
- kStatistics package for R
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